Large deviations for multiple ergodic averages

Abstract

The main purpose of this work is to estimate how multiple ergodic averages appart from a given quantity. This problem can be studied by describing a large deviation process for empirical measures as obtained by using the contraction principle. The case of single ergodic averages for empirical measures was already studied by Pfister and Sullivan [Nonlinarity, 10 (2005) 237-261]. To have a more complete picture on empirical measures and V– statistics, we estimate the size of the sets GK = {x : Lr (x) ⊂ K }, where Lr(x) is the limit-point set of the sequence of empirical measures and K is a compact subset of ℳ(Xr) with ℳ(X) the set of measures on X. In pasrticular, we obtain a variational formula for the topological entropy of Gk. The result of this work about the dimension of the sets Gk can be compared with the one recently circulated by Fan, Schemeling and Wu [arXiv:1206.3214v1 (2012)].